Newform orbit 156.3.v.a

• Label: 156.3.v.a
• Level: $156$
• Weight: $3$
• Character orbit: 156.v
• Analytic conductor: $4.251$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	156.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.25069212402\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(52\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 12 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.99724	−	0.105010i	2.32452	+	1.89647i	3.97795	+	0.419462i	−4.07052	+	4.07052i	−4.44349	−	4.03180i	−6.36237	+	1.70479i	−7.90087	−	1.25549i	1.80683	+	8.81677i	8.55726	−	7.70236i
11.2	−1.99486	+	0.143268i	−0.996699	+	2.82959i	3.95895	−	0.571598i	0.583261	−	0.583261i	1.58289	−	5.78744i	9.99247	−	2.67747i	−7.81566	+	1.70745i	−7.01318	−	5.64050i	−1.07996	+	1.24709i
11.3	−1.98185	−	0.268819i	2.90425	−	0.751900i	3.85547	+	1.06552i	2.56976	−	2.56976i	−5.95791	+	0.709437i	4.17972	−	1.11995i	−7.35454	−	3.14813i	7.86929	−	4.36741i	−5.78367	+	4.40207i
11.4	−1.97039	+	0.342851i	−1.79739	−	2.40195i	3.76491	−	1.35110i	6.28105	−	6.28105i	4.36509	+	4.11656i	2.83339	−	0.759204i	−6.95512	+	3.95301i	−2.53875	+	8.63451i	−10.2227	+	14.5296i
11.5	−1.92803	+	0.531690i	0.202664	−	2.99315i	3.43461	−	2.05023i	−3.22527	+	3.22527i	1.20068	+	5.87864i	−6.48508	+	1.73767i	−5.53196	+	5.77905i	−8.91785	−	1.21321i	4.50359	−	7.93328i
11.6	−1.88718	+	0.662233i	−2.90391	+	0.753198i	3.12289	−	2.49951i	−4.53056	+	4.53056i	4.98141	−	3.34449i	−2.66256	+	0.713431i	−4.23821	+	6.78510i	7.86539	−	4.37444i	5.54969	−	11.5503i
11.7	−1.85074	−	0.758122i	−2.90425	+	0.751900i	2.85050	+	2.80618i	2.56976	−	2.56976i	5.94505	+	0.810197i	−4.17972	+	1.11995i	−3.14813	−	7.35454i	7.86929	−	4.36741i	−6.70414	+	2.80777i
11.8	−1.78217	−	0.907679i	−2.32452	−	1.89647i	2.35224	+	3.23527i	−4.07052	+	4.07052i	2.42131	+	5.48974i	6.36237	−	1.70479i	−1.25549	−	7.90087i	1.80683	+	8.81677i	10.9491	−	3.55962i
11.9	−1.66211	+	1.11238i	−0.532038	+	2.95245i	1.52523	−	3.69779i	4.57676	−	4.57676i	−2.39993	−	5.49912i	−12.3258	+	3.30269i	1.57823	+	7.84278i	−8.43387	−	3.14163i	−2.51600	+	12.6982i
11.10	−1.65597	−	1.12150i	0.996699	−	2.82959i	1.48446	+	3.71435i	0.583261	−	0.583261i	−4.82390	+	3.56791i	−9.99247	+	2.67747i	1.70745	−	7.81566i	−7.01318	−	5.64050i	−1.61999	+	0.311732i
11.11	−1.53499	−	1.28211i	1.79739	+	2.40195i	0.712363	+	3.93606i	6.28105	−	6.28105i	0.320605	−	5.99143i	−2.83339	+	0.759204i	3.95301	−	6.95512i	−2.53875	+	8.63451i	−17.6943	+	1.58830i
11.12	−1.53298	+	1.28451i	2.21038	−	2.02836i	0.700047	−	3.93827i	1.75181	−	1.75181i	−0.783012	+	5.94869i	0.381687	−	0.102273i	3.98560	+	6.93650i	0.771546	−	8.96687i	−0.435262	+	4.93571i
11.13	−1.51316	+	1.30780i	2.30367	+	1.92174i	0.579330	−	3.95782i	−0.238613	+	0.238613i	−5.99908	+	0.104830i	5.00111	−	1.34004i	4.29941	+	6.74648i	1.61382	+	8.85413i	0.0490031	−	0.673119i
11.14	−1.40388	−	1.42447i	−0.202664	+	2.99315i	−0.0582443	+	3.99958i	−3.22527	+	3.22527i	4.54817	−	3.91333i	6.48508	−	1.73767i	5.77905	−	5.53196i	−8.91785	−	1.21321i	9.12221	+	0.0664181i
11.15	−1.30323	−	1.51710i	2.90391	−	0.753198i	−0.603188	+	3.95426i	−4.53056	+	4.53056i	−4.92714	−	3.42393i	2.66256	−	0.713431i	6.78510	−	4.23821i	7.86539	−	4.37444i	12.7777	+	0.968957i
11.16	−1.25480	+	1.55739i	−1.93783	−	2.29015i	−0.850952	−	3.90844i	−2.14320	+	2.14320i	5.99826	−	0.144281i	7.70127	−	2.06355i	7.15475	+	3.57904i	−1.48961	+	8.87587i	−0.648518	−	6.02709i
11.17	−0.889190	+	1.79146i	−2.91776	−	0.697637i	−2.41868	−	3.18590i	1.82920	−	1.82920i	3.84423	−	4.60672i	−8.86485	+	2.37533i	7.85810	−	1.50011i	8.02660	+	4.07107i	1.65044	+	4.90346i
11.18	−0.883243	−	1.79440i	0.532038	−	2.95245i	−2.43976	+	3.16979i	4.57676	−	4.57676i	−5.76780	+	1.65304i	12.3258	−	3.30269i	7.84278	+	1.57823i	−8.43387	−	3.14163i	−12.2549	−	4.17016i
11.19	−0.832283	+	1.81860i	−1.04033	+	2.81384i	−2.61461	−	3.02718i	−5.86854	+	5.86854i	−4.25140	−	4.23386i	0.482970	−	0.129411i	7.68132	−	2.23546i	−6.83541	−	5.85467i	−5.78824	−	15.5568i
11.20	−0.685341	−	1.87891i	−2.21038	+	2.02836i	−3.06061	+	2.57539i	1.75181	−	1.75181i	5.32596	+	2.76299i	−0.381687	+	0.102273i	6.93650	+	3.98560i	0.771546	−	8.96687i	−4.49208	−	2.09090i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
13.f	odd	12	1	inner
39.k	even	12	1	inner
52.l	even	12	1	inner
156.v	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	156.3.v.a	✓	208
3.b	odd	2	1	inner	156.3.v.a	✓	208
4.b	odd	2	1	inner	156.3.v.a	✓	208
12.b	even	2	1	inner	156.3.v.a	✓	208
13.f	odd	12	1	inner	156.3.v.a	✓	208
39.k	even	12	1	inner	156.3.v.a	✓	208
52.l	even	12	1	inner	156.3.v.a	✓	208
156.v	odd	12	1	inner	156.3.v.a	✓	208

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.3.v.a	✓	208	1.a	even	1	1	trivial
156.3.v.a	✓	208	3.b	odd	2	1	inner
156.3.v.a	✓	208	4.b	odd	2	1	inner
156.3.v.a	✓	208	12.b	even	2	1	inner
156.3.v.a	✓	208	13.f	odd	12	1	inner
156.3.v.a	✓	208	39.k	even	12	1	inner
156.3.v.a	✓	208	52.l	even	12	1	inner
156.3.v.a	✓	208	156.v	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(156, [\chi])\).